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Space-Varying Iterative Restoration of Diffuse Optical Tomograms Reconstructed by the Photon Average Trajectories Method
EURASIP Journal on Advances in Signal Processing volume 2007, Article number: 034747 (2007)
The possibility of improving the spatial resolution of diffuse optical tomograms reconstructed by the photon average trajectories (PAT) method is substantiated. The PAT method recently presented by us is based on a concept of an average statistical trajectory for transfer of light energy, the photon average trajectory (PAT). The inverse problem of diffuse optical tomography is reduced to a solution of an integral equation with integration along a conditional PAT. As a result, the conventional algorithms of projection computed tomography can be used for fast reconstruction of diffuse optical images. The shortcoming of the PAT method is that it reconstructs the images blurred due to averaging over spatial distributions of photons which form the signal measured by the receiver. To improve the resolution, we apply a spatially variant blur model based on an interpolation of the spatially invariant point spread functions simulated for the different small subregions of the image domain. Two iterative algorithms for solving a system of linear algebraic equations, the conjugate gradient algorithm for least squares problem and the modified residual norm steepest descent algorithm, are used for deblurring. It is shown that a gain in spatial resolution can be obtained.
Arridge SR: Optical tomography in medical imaging. Inverse Problems 1999,15(2):R41-R93. 10.1088/0266-5611/15/2/022
Yodh A, Chance B: Spectroscopy and imaging with diffusing light. Physics Today 1995,48(3):34-40. 10.1063/1.881445
Hielscher AH, Klose AD, Hanson KM: Gradient-based iterative image reconstruction scheme for time-resolved optical tomography. IEEE Transactions on Medical Imaging 1999,18(3):262-271. 10.1109/42.764902
Lyubimov VV: Physical foundations of the strongly scattering media laser tomography. Laser Optics '95: Biomedical Applications of Lasers, June 1996, St. Petersburg, Russia, Proceedings of SPIE 2769: 107–110.
Lyubimov VV: Optical tomography of highly scattering media by using the first transmitted photons of ultrashort pulses. Optics and Spectroscopy 1996,80(4):616-619.
Lyubimov VV, Mironov EP, Murzin AG, Volkonsky VB, Kravtsenyuk OV: Calculation of shadows induced by macroinhomogeneities located inside a strongly scattering object using the integration over the average photon path. Photon Propagation in Tissues III, September 1997, San Remo, Italy, Proceedings of SPIE 3194: 409–416.
Lyubimov VV: On the spatial resolution of optical tomography of strongly scattering media with the use of the directly passing photons. Optics and Spectroscopy 1999,86(2):251-252.
Volkonskii VB, Kravtsenyuk OV, Lyubimov VV, Mironov EP, Murzin AG: The use of statistical characteristics of photon trajectories for the tomographic studies of optical macroheterogeneities in strongly scattering objects. Optics and Spectroscopy 1999,86(2):253-260.
Kravtsenyuk OV, Lyubimov VV: Specific features of statistical characteristics of photon trajectories in a strongly scattering medium near an object surface. Optics and Spectroscopy 2000,88(4):608-614. 10.1134/1.626846
Kravtsenyuk OV, Lyubimov VV: Application of the method of smooth perturbations to the solution of problems of optical tomography of strongly scattering objects containing absorbing macroinhomogeneities. Optics and Spectroscopy 2000,89(1):107-112. 10.1134/1.1131523
Lyubimov VV, Kalintsev AG, Konovalov AB, et al.: Application of the photon average trajectories method to real-time reconstruction of tissue inhomogeneities in diffuse optical tomography of strongly scattering media. Physics in Medicine and Biology 2002,47(12):2109-2128. 10.1088/0031-9155/47/12/308
Lyubimov VV, Konovalov AB, Kutuzov II, et al.: Influence of fast reconstruction algorithms on spatial resolution of optical diffuse tomography by photon average trajectories method. Saratov Fall Meeting 2001: Optical Technologies in Biophysics and Medicine III, October 2002, Saratov, Russia, Proceedings of SPIE 4707: 53–59.
Konovalov AB, Lyubimov VV, Kutuzov II, et al.: Application of integral transform algorithms to high-resolution reconstruction of tissue inhomogeneities in medical diffuse optical tomography. Optics in Health Care and Biomedical Optics: Diagnostics and Treatment, October 2002, Shanghai, China, Proceedings of SPIE 4916: 9–21.
Konovalov AB, Lyubimov VV, Kutuzov II, et al.: Application of transform algorithms to high-resolution image reconstruction in optical diffusion tomography of strongly scattering media. Journal of Electronic Imaging 2003,12(4):602-612. 10.1117/1.1604119
Lyubimov VV, Kravtsenyuk OV, Kalintsev AG, et al.: The possibility of increasing the spatial resolution in diffusion optical tomography. Journal of Optical Technology 2003,70(10):715-720. 10.1364/JOT.70.000715
Golubkina OV, Kalintsev AG, Konovalov AB, et al.: Application of photon average trajectories method for separate mapping of absorbing and scattering macroinhomogeneities using time-domain measurements technique. Photon Migration, Optical Coherence Tomography, and Microscopy, June 2001, Munich, Germany, Proceedings of SPIE 4431: 275–281.
Nagy JG, Palmer K, Perrone L: Iterative methods for image deblurring: a Matlab object-oriented approach. Numerical Algorithms 2004,36(1):73-93.
Nagy JG, O'Leary DP: Restoring images degraded by spatially variant blur. SIAM Journal of Scientific Computing 1998,19(4):1063-1082. 10.1137/S106482759528507X
Nagy JG, O'Leary DP: Fast iterative image restoration with a spatially-varying PSF. Advanced Signal Processing: Algorithms, Architectures, and Implementations VII, July 1997, San Diego, Calif, USA , Proceedings of SPIE 3162: 388–399.
Björck Å: Numerical Methods for Least Squares Problems. SIAM, Philadelphia, Pa, USA; 1996.
Kaufman L: Maximum likelihood, least squares, and penalized least squares for PET. IEEE Transactions on Medical Imaging 1993,12(2):200-214. 10.1109/42.232249
Nagy JG, Strakos Z: Enforcing nonnegativity in image reconstruction algorithms. Mathematical Modeling, Estimation, and Imaging, July 2000, San Diego, Calif, USA, Proceedings of SPIE 4121: 182–190.
Schweiger M, Arridge SR, Hiraoka M, Delpy DT: The finite element method for the propagation of light in scattering media: boundary and source conditions. Medical Physics 1995,22(11):1779-1792. 10.1118/1.597634
Sandwell DT: Biharmonic spline interpolation of GEOS-3 and SEASAT altimeter data. Geophysical Research Letters 1987,14(2):139-142. 10.1029/GL014i002p00139
Kak AC, Slaney M: Principles of Computerized Tomographic Imaging. IEEE Press, New York, NY, USA; 1988.
The Math Work : Using Matlab, Version 6. 2000.
Papoulis A: Systems and Transforms with Applications in Optics. McGraw-Hills, New York, NY, USA; 1968.
Feng S, Zeng F, Chance B: Monte Carlo simulations of photon migration path distributions in multiple scattering media. Photon Migration and Imaging in Random Media and Tissues, January 1993, Los Angeles, Calif, USA, Proceedings of SPIE 1888: 78–89.
McNown SR, Jain AK: Approximate shift-invariance by warping shift-variant systems. In The Restoration of HST Images and Spectra II. Edited by: Hanisch RJ, White RL. Space Telescope Science Institute, Baltimore, Md, USA; 1994:181-187.
Robbins GM, Huang TS: Inverse filtering for linear shift-variant imaging systems. Proceedings of the IEEE 1972,60(7):862-872.
Sawchuk AA: Space-variant image restoration by coordinate transformations. Journal of the Optical Society of America 1974,64(2):138-144. 10.1364/JOSA.64.000138
Adorf H-M: Towards HST restoration with space-variant PSF, cosmic rays and other missing data. In The Restoration of HST Images and Spectra II. Edited by: Hanisch RJ, White RL. Space Telescope Science Institute, Baltimore, Md, USA; 1994:72-78.
Trussell HJ, Fogel S: Identification and restoration of spatially variant motion blurs in sequential images. IEEE Transactions on Image Processing 1992,1(1):123-126. 10.1109/83.128039
Fish DA, Grochmalicki J, Pike ER: Scanning singular-value-decomposition method for restoration of images with space-variant blur. Journal of the Optical Society of America A: Optics, Image Science, and Vision 1996,13(3):464-469. 10.1364/JOSAA.13.000464
Kamm J, Nagy JG: Kronecker product and SVD approximations in image restoration. Linear Algebra and Its Applications 1998,284(1–3):177-192.
Restore Tools: An Object Oriented Matlab Package for Image Restoration 2002.https://doi.org/www.mathcs.emory.edu/~nagy/RestoreTools
Hanke M: Conjugate Gradient Type Methods for Ill-Posed Problems, Pitman Research Notes in Mathematics. Longman Scientific & Technical, Harlow, UK; 1995.
Nagy JG, Palmer KM: Steepest descent, CG, and iterative regularization of ill-posed problems. BIT Numerical Mathematics 2003,43(5):1003-1017.
Vogel CR: Computational Methods for Inverse Problems. SIAM, Philadelphia, Pa, USA; 2002.
Bertero M, Boccacci P: Introduction to Inverse Problems in Imaging. IOP, London, UK; 1998.
Richardson WH: Bayesian-based iterative method of image restoration. Journal of the Optical Society of America 1972,62(1):55-59. 10.1364/JOSA.62.000055
Lucy LB: An iterative technique for the rectification of observed distributions. The Astronomical Journal 1974,79(6):745-753.
Golub GH, van Loan CF: Matrix Computations. 3rd edition. John Hopkins Institute Press, Baltimore, Md, USA; 1989.
Tsui BMW, Zhao X, Frey EC, Gullberg GT: Comparison between ML-EM and WLS-CG algorithms for SPECT image reconstruction. IEEE Transactions on Nuclear Science 1991,38(6, part 2):1766-1772.
Jiang M, Wang G, Skinner MW, Rubinstein JT, Vannier MW: Blind deblurring of spiral CT images. IEEE Transactions on Medical Imaging 2003,22(7):837-845. 10.1109/TMI.2003.815075
Groetsch CW: The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind. Pitman, Boston, Mass, USA; 1984.
Hansen PC, O'Leary DP: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing 1993,14(6):1487-1503. 10.1137/0914086
Kilmer ME, O'Leary DP: Choosing regularization parameters in iterative methods for ill-posed problems. SIAM Journal on Matrix Analysis and Applications 2001,22(4):1204-1221. 10.1137/S0895479899345960
Calvetti D, Landi G, Reichel L, Sgallari F: Non-negativity and iterative methods for ill-posed problems. Inverse Problems 2004,20(6):1747-1758. 10.1088/0266-5611/20/6/003
Dehghani H, Pogue BW, Poplack SP, Paulsen KD: Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results. Applied Optics 2003,42(1):135-145. 10.1364/AO.42.000135
Li A, Miller EL, Kilmer ME, et al.: Tomographic optical breast imaging guided by three-dimensional mammography. Applied Optics 2003,42(25):5181-5190. 10.1364/AO.42.005181
Hebden JC, Arridge SR, Schweiger M: Investigation of alternative data types for time resolved optical tomography. OSA Technical Digest, Biomedical Topical Meetings, 1998, Washington, DC, USA 21: 162–167.
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Konovalov, A.B., Vlasov, V.V., Kravtsenyuk, O.V. et al. Space-Varying Iterative Restoration of Diffuse Optical Tomograms Reconstructed by the Photon Average Trajectories Method. EURASIP J. Adv. Signal Process. 2007, 034747 (2007). https://doi.org/10.1155/2007/34747
- Steep Descent
- Point Spread Function
- Linear Algebraic Equation
- Gradient Algorithm